A summetria is a collection or an experience of relationships that make individual elements or entities into a group.
There are bonds between the entites, bindings of the elements and a boundary common to all elements of a group, and thes bonds, bindings anf boundaies are the summetria of the group
A symmetry is one type of summetria in which the bonds and bindings and the boundary areapprehended inerms of equality of measure, or a common measure, under rotation.
The Greek Iso mans eveness of rotation and isometries therefore are invariant "statuses" of form under rotation . Sym or sum means common or shared, but the measurement that is shared is usually not specifically defined unless rotation , translation or reflection is invoked. The key idea is Invariance after Action.
Summetria is more general than that notion, but as a defining notion symmetry forms groups. Different symmetries form different groups, and thus symmetry is a group defining Characterisitic.
Both Justus and Hermann used these otions of symmetry in a group defining way, and Hermann in particular sought the invariant form as a symmetrical object.
The Audehnungslehre revels in and advances the importance of symmetry in theory design, notation design and terminolgy, and as a consisent set of Analytical principles proposed symmetry as a fundamental indication of natural balance.
At the same time as philosophers abd scientists were taking note of the natural symmetry nature produced as an indication of a deeper simpler order, the empirical field effect was being proposed. Scientists did not view space as empty, but rather full of Aetheric substance. Thus behaviours resulted from this fluidic motion of aether.
When it became impossible to verify his aether, the observed field effects had to be rationalised in some way. The shift was from some invisible field in space to the behaviour of the observable particles as a group. As a group , particles at first could be distinguished, but at smaller and smaller particles it became difficult to distinguish a single particle from a group of symmetrical particles: the y all behaved in the same way.
Thus by concentrating the attention on groups of symmetries a more general picture emerged for particle behaviour, and the field, while not displaced as the cause was rather rigorously defined in terms of the group symmetries. Thus, given any collection of particles, if a group symmetry could be found, that was taken as sufficient evidence that a field existed between those particles causative of that behaviour.
So srong was this idea that symmetries were organising principles that theoretical Physicists took it to the next level and predicted certain behaviours and techniques which preserved symmetries, and consequently allowed analogous thinking to solve difficult problems.
The principle of symmetry and invariance is a Grassmanian concern which has influenced modern physics inso many ways, but is the fractal Geometry of Benoit Mandelbrot which makes it all work together naturally. Because of the notion of self similarity(almost) at every level a set of symmetries can now form the networks of a field at every level.
Now at every level it is not necessarily identical to the first level, thus gradually over levels the symmetry link or bond breaks down, and new symmetries may form. This behaviour is fractal, and liked to developments over many scales.
The field effect concepts have attempted to keep pace with the freedom symmetry groups are used to build models of reality, but at the end it requires geometrical notions to guide our subjective processing into symmetrical ways of analogising observed behaviour to give insight into the combinatorial forms of Fractal Space.
The Intensive workings of lower levels of fractal geometry are equally important as the extensive magnitudes of the Ausdehnungs groesse.
In "free space objects spiral and twist in open or closed curves because the symmetries are not exact at every level of scale, thus the fields cause rotation, condensation and compaction as well as expansion and extraction.